Energy method for computing periodic solutions of strongly nonlinear systems (I) — Autonomous systems

In this paper a new method for solving for the periodic solution (limit cycle) of a strongly nonlinear system is suggested. Using this method not only the existence, stability and number of periodic solutions can be decided, but at the same time the approximate expressions for these periodic solutions can also be obtained. The proof of this method is given as well.The project is supported by the National Natural Science Foundation of China.     

Generalized hyperbolic perturbation method for homoclinic solutions of strongly nonlinear autonomous systems

A generalized hyperbolic perturbation method is presented for homoclinic solutions of strongly nonlinear autonomous oscillators,in which the perturbation procedure is improved for those systems whose exact homoclinic generating solutions cannot be explicitly derived.The generalized hyperbolic functions are employed as the basis functions in the present procedure to extend the validity of the hyperbolic perturbation method.Several strongly nonlinear oscillators with quadratic,cubic,and quartic nonlinearity are studied in detail to illustrate the efficiency and accuracy of the present method.     

On numerical solutions of periodically perturbed conservative systems

A nonlinear perturbed conservative system is discussed. By means of Hadamard's theorem, the existence and uniqueness of the solution of the continuous problem are proved. When the equation is discreted on the uniform meshes, it is proved that the corresponding discrete problem has a unique solution. Finally, the accuracy of the numerical solution is considered and a simple algorithm is provided for solving the nonlinear difference equation. Communicated by SU Yu-cheng Biography: LIU Guo-qing (1966-)     

Dirac reduction of binary asteroid systems with symmetry and stability of relative equilibria

In this paper, we study symmetry reduction for a binary asteroid system modeled by a rigid body and a particle. In particular, we demonstrate how translational and rotational symmetry reduction appeared in the binary asteroid system can be carried out in the context of Dirac reduction by stages and with the associated reduction of implicit Hamiltonian systems. Then we investigate stability of relative equilibria of the asteroid pair and show stability regions by using the energy-momentum method. Lastly, we illustrate some numerical simulations for stable and unstable orbits near from relative equilibria of the Collinear and T configurations.     

An AFVS model for polymer solutions: New scaling relationship

The altered free volume state (AFVS) approach developed by Kulkarni and Mashelkar has been extended to develop a model for the viscosity of polymer solutions. The effect of polymer concentration, molecular weight and electrolyte concentration is shown to be satisfactorily explained in the framework of AFVS model. The regimes of viscosity behaviour have been defined for the systems investigated and the transitions involved identified. The approach leads to the identification of a surprisingly simple and unique scaling parameterΦ _P, which enables excellent concentration-molecular weight superposition.     

On the existence of solutions of boundary value problems of duffing type systems

ＩｎｔｒｏｄｕｃｔｉｏｎＩｎ [1 ] ,ｕｎｄｅｒａｇｒｏｕｐｏｆｖｅｒｙｅｘｔｅｎｓｉｖｅｃｏｎｄｉｔｉｏｎｓ,ＳＨＥＮＺｕ＿ｈｅｓｔｕｄｉｅｄｔｈｅｅｘｉｓｔｅｎｃｅｏｆａｕｎｉｑｕｅ 2π＿ｐｅｒｉｏｄｉｃｓｏｌｕｔｉｏｎｏｆｔｈｅｓｙｓｔｅｍｏｆｏｒｄｉｎａｒｙｄｉｆｆｅｒｅｎｔｉａｌｅｑｕａｔｉｏｎｓｕ″(ｔ) Ｇ(ｕ(ｔ) ) =ｐ(ｔ) ,(1 )ｗｈｅｒｅＧ :Ｒｎ →Ｒｈａｓａｃｏｎｔｉｎｕｏｕｓｓｅｃｏｎｄｐａｒｔｉａｌｄｅｒｉｖａｔｉｖｅｓ,ａｎｄｐ:Ｒ→Ｒｎｉｓｃｏｎｔｉｎｕｏｕｓａｎｄ2π＿ｐｅｒ…     

On a new algorithm of constructing solitary wave solutions for systems of nonlinear evolution equations in mathematical physics

ＩｎｔｒｏｄｕｃｔｉｏｎＳｏｌｖｉｎｇｎｏｎｌｉｎｅａｒｅｑｕａｔｉｏｎｓｉｓａｌｗａｙｓａｖｅｒｙｉｎｔｅｒｅｓｔｉｎｇｓｕｂｊｅｃｔｆｏｒｍａｔｈｅｍａｔｉｃｉａｎａｎｄｐｈｙｓｉｃｉａｎ ,ｉｎｐａｒｔｉｃｕｌａｒ,ｓｏｌｉｔａｒｙｗａｖｅｓｏｌｕｔｉｏｎｓｆｏｒｎｏｎｌｉｎｅａｒｅｑｕａｔｉｏｎｓａｒｅｏｆｂｏｔｈｔｈｅｏｒｅｔｉｃａｌａｎｄｐｒａｃｔｉｃａｌｉｍｐｏｒｔａｎｃｅ.Ｒｅｃｅｎｔｌｙ ,Ｙａｎ[1]ｏｂｔａｉｎｅｄａｔｒａｎｓｆｏｒｍａｔｉｏｎｄｉｒｅｃｔｌｙｆｒｏｍｔｈｅｆａｍｏ…     

Nontrivial equilibrium solutions for a semilinear reaction-diffusion system

By the degree theory on positive cone together with the technique of a priori estimate, the nontrivial equilibrium solutions of a strong nonlinearity and weak coupling reaction diffusion system and the structure of the equilibrium solutions are discussed.     

Understanding thixotropic and antithixotropic behavior of viscoelastic micellar solutions and liquid crystalline dispersions. I. The model

A simple model consisting of the Upper Convected Maxwell constitutive equation and a kinetic equation for destruction and construction of structure, first proposed by Fredrickson in 1970, is used here to reproduce the complex rheological behavior of viscoelastic systems that also exhibit thixotropy and rheopexy under shear flow. The model requires five parameters that have physical significance and that can be estimated from rheological measurements. Several steady and unsteady flow situations were analyzed with the model. The model predicts creep behavior, stress relaxation and the presence of thixotropic loops when the sample is subjected to transient stress cycles. Such behavior has been observed with surfactant-based solutions and dispersions. The role of the characteristic time for structure built up, λ, in the extent and shape of the thixotropic loops is demonstrated.     

On the asymptotical behaviour of solutions of a class of nonlinear control systems

Ｃｏｎｓｉｄｅｒｔｈｅｆｏｌｌｏｗｉｎｇｎｏｎｌｉｎｅａｒｃｏｎｔｒｏｌｓｙｓｔｅｍ：·ｘ＝Ａｘ＋Ｂ（σ）,σ＝ＣＴｘ,（１）ｗｈｅｒｅｘ＝（ｘ１,ｘ２,…,ｘｎ）Ｔ∈Ｒｎ,ｔｈｅｎｏｔｅＴｅｘｐｒｅｓｓｅｓｔｈｅｔｒａｎｓｐｏｓｅｏｆｍａｔｒｉｘａｎｄｖｅｃｔｏｒ,Ａ,ＢａｎｄＣａｒｅｎ×ｎ,ｎ×ｍａｎｄｎ×ｍｃｏｎｓｔａｎｔｍａｔｒｉｃｅｓｒｅｓｐｅｃｔｉｖｅｌｙ,σ＝（σ１,σ２,…,σｍ）Ｔ∈Ｒｍａｎｄ（σ）＝（１（σ１）,２（σ２）,…,ｍ（σｍ））Ｔ：Ｒｍ→Ｒｍｉｓｃｏｎｔ…     

Uniformly ultimate boundedness for discontinuous systems with time-delay

The uniformly ultimate boundedness of discontinuous systems with time-delay in the sense of Filippov solutions is discussed. Based on the Lyapunov-Krasovskii functional, the Lyapunov theorem for the globally strongly uniformly ultimate boundedness of retarded discontinuous systems is presented. Furthermore, the result is applied to a class of mechanical systems with a retarded discontinuous friction item.     

Bifurcations of traveling wave solutions and exact solutions to generalized Zakharov equation and Ginzburg-Landau equation

This paper studies the dynamic behaviors of some exact traveling wave solutions to the generalized Zakharov equation and the Ginzburg-Landau equation. The effects of the behaviors on the parameters of the systems are also studied by using a dynamical system method. Six exact explicit parametric representations of the traveling wave solutions to the two equations are given.     

一类非自治动力系统超次谐周期解的不存在性

在非线性动力系统的研究中， Melnikov函数被广泛地用来作为微扰哈密顿系统是否发生次谐或超次谐分岔乃至混沌的判 据. 但是在大多数情况下，经典的Melnikov方法往往只给出存在次谐周期解的结论. 产生 该结果的原因被归之为在经典的Melnikov方法中只采取了一阶近似，因而高阶Melnikov方 法被发展用来判断超次谐周期解的存在性. 本文对一类非自治微分动力系统进行了研究，证 明了在这样一类系统中如果存在周期解则只可能是次谐周期解，超次谐周期解不可能存在， 并进一步证明了在一类平面问题中所定义的旋转(R)型超次谐周期解同样不可能存在.作为 该结论的一个应用，文中考察了几个典型的算例，结果表明现有的二阶Melnikov方法判断 平面扰动系统是否存在超次谐周期解的结论是不恰当的，并提供了一个简单的几何上的解释.     

Finite element analysis of flow,heat transfer,and free interfaces in an electron-beam vaporization system for metals

A numerical analysis is made of the liquid flow and energy transport in a system to evaporate metals. The energy from an electron-beam heats an axisymmetric metal disk supported by a water-cooled platform. Metal evaporates from the surface of a hot pool of liquid which is surrounded by a shell of its own solid. Flow in the pool is strongly driven by temperature-induced buoyancy and capillary forces, and is located in the transition region between laminar and turbulent flow. The evaporation rate is strongly influenced by the locations of the free boundaries. A modified finite element method is used to calculate the steady state flow and temperature fields coupled with the interface locations. The mesh is structured with spines that stretch and pivot as the interfaces move. The discretized equations are arranged in an ‘arrow’ matrix and are solved using the Newton–Raphson method. The electron-beam power and platform contact resistance are varied for cases involving the evaporation of aluminum. The results reveal the interaction of liquid flow, heat transfer and free interfaces. © 1998 John Wiley & Sons, Ltd.     

The functional similarity between faxén relations and singularity solutions for fluid-fluid, fluid-solid and solid-solid dispersions

The functional similarity between Faxén relations (for moments of the appropriate transport flux) and singularity solutions has been noted in the past. For rigid particles (and perfect conductors etc.) it has been noted in the literature that the root of this similarity is linked to the Lorentz reciprocal theorem. However, the duality applies even to more general two-phase problems such as a viscous drop in another solvent, with the relevant singularity distribution taken from the exterior solution. Although two-phase Faxén relations are available for various particle shapes, until now, the root of this duality has not been demonstrated explicitly. The application of the duality is illustrated by the derivation of new Faxén relations for ellipsoidal inclusions.     

The MLP method for subharmonic and ultraharmonic resonance solutions of strongly nonlinear systems

ＩｎｔｒｏｄｕｃｔｉｏｎＴｈｅｓｔｕｄｙｏｆｔｈｅａｐｐｒｏｘｉｍａｔｅｓｏｌｕｔｉｏｎｏｆｓｔｒｏｎｇｌｙｎｏｎｌｉｎｅａｒｓｙｓｔｅｍｓｉｓｓｔｉｌｌｔｈｅｍａｉｎｔａｓｋｉｎｔｈｅｎｏｎｌｉｎｅａｒｄｙｎａｍｉｃａｌｒｅｇｉｏｎ .ＴｈｅｒｅａｒｅｍａｔｕｒｅｍｅｔｈｏｄｓｉｎｃｌｕｄｉｎｇｔｈｅｄｅｖｅｌｏｐｅｄＫＢＭｍｅｔｈｏｄ[1],ｔｈｅｍｏｄｉｆｉｅｄＬＰｍｅｔｈｏｄ[2 ]ａｎｄｔｈｅｓｔｒｏｂｏｓｃｏｐｉｃｍｅｔｈｏｄ[3]ｅｔａｌ.Ｈｏｗｅｖｅｒ,ｗｈｅｎｔｈｅｓｅｍｅｔｈｏｄｓａｒｅａ…     

Limiting analytical solutions for complex saturated and unsaturated transport problems

Non-linear diffusion and velocity-dependent dispersion problems are under consideration. The necessary and sufficient conditions allowing the comparison of solutions to the two dimensional convection-dispersion equations with different coefficients are obtained. These conditions provide a framework within which solutions to the complex non-linear problems mentioned above can be estimated by solutions to the problems possessing analytical solvability.Nomenclature c(x, y, t) concentration of solute in solution,ML ^–3 - C(h)=d/dh moisture capacity function - D,D _ij hydrodynamic dispersion coefficient, a second order tensor,L ² T ^–1 - D _L longitudinal hydrodynamic dispersion coefficient,L ² T ^–1 - D _m molecular diffusion coefficient,L ² T ^–1 - D _T transverse hydrodynamic coefficient,L ² T ^–1 - G flow domain for the unsaturated flow problem - G _z , G _w flow domain and complex potential domain, respectively, for the hydrodynamic dispersion problem - h piezometric head,L - I _n given mass flux normal to the boundary,MLT ^–1 - k hydraulic conductivity,LT ^–1 - K(h) unsaturated hydraulic conductivity,LT ^–1 - L continuously differentiable function with respect to all arguments - m porosity - n(x,t) outer normal vector to the boundary - t time,T - V(x, y, t) seepage velocity vector withV=V,LT ^–1 - x Cartesian coordinate system - x horizontal coordinate,L - y vertical coordinate (elevation),L - (x),(x,t) given functions in initial and boundary conditions (3), (4) - ₁(,) angle between vectors ₁c andV - boundary of the flow domain - _L , _T longitudinal and transverse dispersivities, respectively,L - water mass density,ML ^–3 - v _i components of a unit vector in the direction of the outward normal to the boundary - =–kh velocity potential - =/m - stream function defined such thatw=+i is the complex potential - =/m     

可燃液体爆燃特性及其抑制实验

设计了一套可燃液体爆炸特性实验装置,利用该装置和立式爆轰管对RP-5油料、RP-3油料及工业酒精的爆炸特性、1301惰性气体对这3种燃料的抑制进行了研究。结果表明:RP-5油料、RP-3油料及工业酒精爆炸的体积分数范围分别为1.53%～7.73%、0.82%～7.17%及 3.38%～18.25%;酒精云雾爆轰的临界起爆能为2.11 MJ/m2、爆速和爆压分别为1 609 m/s 、1 480 kPa,爆轰波传播的胞格宽度为14.5 mm,长度为16.2 mm。1301惰性气体对RP-5油料、RP-3油料及工业酒精的最小惰化体积分数分别为6.75%、6.8%及 5.56%;二氧化碳和氮气对RP-3油料的最小惰化体积分数分别为45%和49%;1301惰性气体对油料爆炸抑制效果明显好于二氧化碳与氮气。